Simplify The Following Expression As Much As Possible: $ (4+i)-(3-5i)(-2+5i) $A. $ -15-24i $ B. $ 35-24i $ C. $ 35+26i $ D. $ -15+26i $

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Introduction

In mathematics, complex expressions are a crucial part of algebra and are used to represent quantities that have both real and imaginary parts. Simplifying complex expressions is an essential skill that helps us to solve problems in various fields, including physics, engineering, and computer science. In this article, we will simplify the given expression (4+i)βˆ’(3βˆ’5i)(βˆ’2+5i)(4+i)-(3-5i)(-2+5i) and explore the different steps involved in simplifying complex expressions.

Understanding Complex Numbers

Before we dive into simplifying the given expression, let's briefly review the basics of complex numbers. A complex number is a number that can be expressed in the form a+bia+bi, where aa and bb are real numbers and ii is the imaginary unit, which satisfies the equation i2=βˆ’1i^2=-1. The real part of a complex number is aa, and the imaginary part is bb. Complex numbers can be added, subtracted, multiplied, and divided just like real numbers.

Simplifying the Given Expression

Now, let's simplify the given expression (4+i)βˆ’(3βˆ’5i)(βˆ’2+5i)(4+i)-(3-5i)(-2+5i). To simplify this expression, we need to follow the order of operations (PEMDAS):

  1. Distribute the negative sign: The expression can be rewritten as βˆ’(3βˆ’5i)(βˆ’2+5i)-(3-5i)(-2+5i).
  2. Multiply the two complex numbers: To multiply two complex numbers, we need to follow the distributive property. We multiply each term in the first complex number by each term in the second complex number and then combine like terms.

βˆ’(3βˆ’5i)(βˆ’2+5i)=βˆ’((3)(βˆ’2)+(3)(5i)+(βˆ’5i)(βˆ’2)+(βˆ’5i)(5i))-(3-5i)(-2+5i) = -((3)(-2) + (3)(5i) + (-5i)(-2) + (-5i)(5i))

=βˆ’(βˆ’6+15i+10iβˆ’25i2)= -(-6 + 15i + 10i - 25i^2)

=βˆ’(βˆ’6+25i+25)= -(-6 + 25i + 25)

=βˆ’(βˆ’6+25i+25)= -(-6 + 25i + 25)

=βˆ’(βˆ’31+25i)= -(-31 + 25i)

=31βˆ’25i= 31 - 25i

  1. Subtract the two complex numbers: Now that we have simplified the second part of the expression, we can subtract it from the first part.

(4+i)βˆ’(31βˆ’25i)=4+iβˆ’31+25i(4+i) - (31 - 25i) = 4 + i - 31 + 25i

=βˆ’27+26i= -27 + 26i

Conclusion

In this article, we simplified the given expression (4+i)βˆ’(3βˆ’5i)(βˆ’2+5i)(4+i)-(3-5i)(-2+5i) and arrived at the final answer βˆ’27+26i-27 + 26i. We followed the order of operations and used the distributive property to multiply the two complex numbers. We also reviewed the basics of complex numbers and the rules for adding, subtracting, multiplying, and dividing complex numbers.

Answer

The correct answer is D. $ -15+26i $ is incorrect, the correct answer is A. $ -15-24i $ is also incorrect, the correct answer is B. $ 35-24i $ is also incorrect, the correct answer is C. $ 35+26i $ is also incorrect, the correct answer is -27 + 26i.

Final Answer

Introduction

In our previous article, we simplified the complex expression (4+i)βˆ’(3βˆ’5i)(βˆ’2+5i)(4+i)-(3-5i)(-2+5i) and arrived at the final answer βˆ’27+26i-27 + 26i. In this article, we will provide a Q&A guide to help you understand the concepts and techniques involved in simplifying complex expressions.

Q: What is the order of operations for simplifying complex expressions?

A: The order of operations for simplifying complex expressions is the same as for real numbers: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This is often remembered by the acronym PEMDAS.

Q: How do I multiply two complex numbers?

A: To multiply two complex numbers, you need to follow the distributive property. Multiply each term in the first complex number by each term in the second complex number and then combine like terms.

Q: What is the difference between a real number and an imaginary number?

A: A real number is a number that can be expressed without any imaginary part, such as 3 or -4. An imaginary number is a number that can be expressed with an imaginary part, such as 3i or -4i.

Q: How do I add and subtract complex numbers?

A: To add or subtract complex numbers, you need to combine the real parts and the imaginary parts separately. For example, to add 3 + 4i and 2 - 5i, you would get:

(3 + 4i) + (2 - 5i) = (3 + 2) + (4i - 5i) = 5 - i

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number is a complex number with the same real part and the opposite imaginary part. For example, the conjugate of 3 + 4i is 3 - 4i.

Q: How do I use the conjugate to simplify complex expressions?

A: You can use the conjugate to simplify complex expressions by multiplying the numerator and denominator by the conjugate of the denominator. This will eliminate the imaginary part from the denominator.

Q: What are some common mistakes to avoid when simplifying complex expressions?

A: Some common mistakes to avoid when simplifying complex expressions include:

  • Forgetting to follow the order of operations
  • Not combining like terms
  • Not using the distributive property when multiplying complex numbers
  • Not using the conjugate to simplify complex expressions

Conclusion

In this article, we provided a Q&A guide to help you understand the concepts and techniques involved in simplifying complex expressions. We covered topics such as the order of operations, multiplying complex numbers, adding and subtracting complex numbers, conjugates, and common mistakes to avoid.

Final Tips

  • Practice simplifying complex expressions regularly to build your skills and confidence.
  • Use the distributive property and the conjugate to simplify complex expressions.
  • Check your work carefully to avoid mistakes.
  • Seek help from a teacher or tutor if you are struggling with complex expressions.

Common Complex Expression Mistakes

  • Forgetting to follow the order of operations
  • Not combining like terms
  • Not using the distributive property when multiplying complex numbers
  • Not using the conjugate to simplify complex expressions

Common Complex Expression Formulas

  • (a+bi)+(c+di)=(a+c)+(b+d)i(a + bi) + (c + di) = (a + c) + (b + d)i
  • (a+bi)βˆ’(c+di)=(aβˆ’c)+(bβˆ’d)i(a + bi) - (c + di) = (a - c) + (b - d)i
  • (a+bi)(c+di)=(acβˆ’bd)+(ad+bc)i(a + bi)(c + di) = (ac - bd) + (ad + bc)i
  • a+bic+di=(a+bi)(cβˆ’di)(c+di)(cβˆ’di)\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)}

Conclusion

Simplifying complex expressions is an essential skill in mathematics, and with practice and patience, you can master it. Remember to follow the order of operations, use the distributive property and the conjugate, and check your work carefully to avoid mistakes.